Digital Image Processing COSC 6380/4393

Transcription

1 Digital Image Processing COSC 6380/4393 Lecture 6 Sept 6 th, 2017 Pranav Mantini Slides from Dr. Shishir K Shah and Frank (Qingzhong) Liu

2 Today Review Logical Operations on Binary Images Blob Coloring Morphological Operations on Binary Images

3 Review: BINARY IMAGES How do binary images arise? Since binary = bi-valued, the (logical) values 0 or 1 usually indicate the absence or presence of an image property in an associated gray-level image: Points of high or low intensity (brightness) Points where an object is present or absent More abstract properties, such as smooth vs. nonsmooth, etc. Convention - We will make the associations 1 = BLACK 0 = WHITE 3

4 Review: BINARY IMAGE GENERATION Tablet-Based Input: Binary images can derive from simple sensors with binary output Simplest example: tablet, resistive pad, or light pen All pixels initially assigned value 0 : I = [I(i, j)], I(i, j) = 0 for all (i, j) = (row column) When pressure or light is applied at (i 0, j 0 ), the image is assigned the value 1 : I(i 0, j 0 ) = 1 This continues until the user completes the drawing 4

5 Review: Grey Level Binary Image Threshold(T) X8 image white box on black background What is good value of T? Binary image

6 Review: Example: How to find T Determine peaks Choose T between peaks(say average) peak 1 peak 2 T

7 Review: Algorithm Initialize T = K/2 Do Compute μ 1 = E X X < K 2 Compute μ 2 = E X X K 2 Set T = μ 1 + μ 2 2 While μ 1! = 0 & μ 2! = 0 AKA: Expectation Maximization (simple version) 7 bimodal histogram well separated peaks

8 DISCUSSION OF HISTOGRAM TYPES We ll return to the histogram later in the context of quantitative gray-level properties Some general qualitative observations are worth making now Bimodal histograms often imply objects and background of significantly different average brightnesses Bimodal histograms are the easiest to threshold The result of thresholding a bimodal histogram is (ideally) a simple binary image showing object/background separation Examples. Images of Printed type Blood cells in solution Machine parts on an assembly line 8

9 HISTOGRAM TYPES Multi-modal histograms often occur when the image contains different objects of different average brightnesses on a uniform background Flat or level histograms usually imply more complex images, containing detail, non-uniform background, etc. Thresholding rarely gives perfect results Usually, some kind of region correction must be applied 9

10 LOGICAL OPERATIONS ON BINARY IMAGES Assume that we have obtained binary images in some way In these and other diagrams that do not use actual digital binary images, we are not showing the discretization into pixels However, since most images are of sufficient resolution that discretization effects are not noticeable, it does not matter 10

11 THE BASIC LOGICAL OPERATIONS We will use only a few simple logical operations Suppose that X 1,..., X n are binary variables For example, pixels from one or more binary images Here is the notation we will use: Logical Complement: NOT(X 1 ) = complement of X 1 Logical AND: AND(X 1, X 2 ) = X 1 X 2 11

12 LOGICAL OPERATIONS Multi-Variable Logical AND: Logical OR: OR(X1, X2) = X1 X2 12

13 LOGICAL OPERATIONS Multi-Variable Logical OR: 13

14 SIMPLE BOOLEAN ALGEBRA PROPERTIES NOT [NOT(X)] = X X1 X2 X3 = (X1 X2 ) X3 = X1 (X2 X3 ) (Associative Law) X1 X2 X3 = (X1 X2 ) X3 = X1 (X2 X3 ) (Associative Law) X1 X2 = X2 X1 (Commutative Law) X1 X2 = X2 X1 (Commutative Law) (X1 X2 ) X3 = (X1 X3 ) (X2 X3 ) (Distributive Law) (X1 X2 ) X3 = (X1 X3 ) (X2 X3 ) (Distributive Law) NOT(X1 X2 ) = NOT(X1 ) NOT(X2 ) (DeMorgan s Law) NOT(X1 X2 ) = NOT(X1 ) NOT(X2 ) (DeMorgan s Law) 14

15 SIMPLE BOOLEAN ALGEBRA PROPERTIES NOT [NOT(X)] = X X1 X2 X3 = (X1 X2 ) X3 = X1 (X2 X3 ) (Associative Law) X1 X2 X3 = (X1 X2 ) X3 = X1 (X2 X3 ) (Associative Law) X1 X2 = X2 X1 (Commutative Law) X1 X2 = X2 X1 (Commutative Law) (X1 X2 ) X3 = (X1 X3 ) (X2 X3 ) (Distributive Law) (X1 X2 ) X3 = (X1 X3 ) (X2 X3 ) (Distributive Law) NOT(X1 X2 ) = NOT(X1 ) NOT(X2 ) (DeMorgan s Law) NOT(X1 X2 ) = NOT(X1 ) NOT(X2 ) (DeMorgan s Law) 15

16 BOOLEAN ALGEBRA Binary Majority (odd # of variables only) 16

17 BOOLEAN ALGEBRA Multi-Variable Binary Majority: MAJ(X 1, X 2,..., X n ) = 1 if more 1 s than 0 s = 0 if more 0 s than 1 s Comments Any binary operation can be created from NOT, AND, OR -Boolean Algebra is an entire math discipline built on these However, we will restrict ourselves to using NOT, AND, OR, and MAJ in a few simple applications 17

18 LOGICAL OPERATIONS ON IMAGES Let I 1, I 2,..., I n be binary images We define logical operations on images on a pointwise basis The complement of an image: J 1 = NOT( I 1 ) if J 1 (i, j) = NOT[ I 1 (i, j) ] for all (i, j) This reverses the contrast - it creates a binary negative: 18

19 BINARY AND The AND or intersection of two images: J 2 = AND(I 1, I 2 ) = I 1 I 2 if J 2 (i, j) = AND[ I 1 (i, j), I 2 (i, j) ] for all (i, j) Shows the overlap of BLACK regions in I 1 and I 2 19

20 BINARY OR The OR or union of two images: J 3 = OR(I 1, I 2 ) = I 1 I 2 if J 3 (i, j) = OR[ I 1 (i, j), I 2 (i, j) ] for all (i, j) Shows the overlap of the WHITE regions in I 1 and I 2 20

21 BINARY OPERATIONS Comments The usefulness of globally applying AND, OR and MAJ to images is very limited. Later, we will find that AND, OR, and MAJ are very useful when applied to small, local image regions There are exceptions... 21

22 EXAMPLE An assembly-line image inspection system. Similar to many marketed by industry: Objective: Numerically compare the stored image I model and the acquired image I 22

23 EXAMPLE Observe that the object in I has been shifted very slightly 23

24 Logical AND The logical AND conveys the overlap A measurement of the displacement is given by: XOR(I, I model ) = OR{ AND[I model, NOT(I)], AND[NOT(I model ), I ]} 24

25 DISPLACEMENT 25

26 EXAMPLE XOR shows where the displacement errors occur To decide if there is a problem or flaw, the ratio or percentage PERCENT = [# black pixels in XOR(I, I model )] / [# black pixels in I model ] may be compared to a pre-determined tolerance percentage P If PERCENT > P, then the part may be flawed or incorrectly placed 26

27 BLOB COLORING A simple technique for region classification and correction Motivation: Gray-level image thresholding usually produces an imperfect binary image: Extraneous blobs or holes due to noise Extraneous blobs from thresholded objects of little interest Nonuniform object/background surface reflectances 27

28 BLOB COLORING It is usually desired to extract a small number of objects or even a single object by thresholding Blob coloring is a very simple technique for listing all of the blobs or objects in a binary image 28

29 BLOB COLORING b b b b b b b b b b b b

30 BLOB COLORING b b b b b b b b b b b b b

31 BLOB COLORING ALGORITHM For binary image I, define a "region color" array R: R(i, j) = region number of pixel I(i, j) Set R = 0 (all zeros) and k = 1 (k = region number counter) While scanning the image left-to-right and top-to-bottom do if I(i, j) = 1 and I(i, j-1) = 0 and I(i-1, j) = 0 then set R(i, j) = k and k = k + 1; if I(i, j) = 1 and I(i, j-1) = 0 and I(i-1, j) = 1 then set R(i, j) = R(i-1, j); 31

32 BLOB COLORING ALGORITHM (contd.) if I(i, j) = 1 and I(i, j-1) = 1 and I(i-1, j) = 0 then set R(i, j) = R(i, j-1); if I(i, j) = 1 and I(i, j-1) = 1 and I(i-1, j) = 1 then set R(i, j) = R(i-1, j); if R(i, j-1) =/= R(i-1, j) then record R(i, j-1) and R(i-1, j) as equivalent (same color) Distinct integers or "colors" k are assigned to each blob Counting the pixels in each blob (by color) is then simple 32

33 EXAMPLE Using blob coloring "Color" of largest blob: 2 33

34 REMOVING MINOR REGIONS Let m = "color" of largest region While scanning the image left-to-right and topto-bottom do if I(i, j) = 1 and R(i, j)!= m then set I(i, j) = 0; 34

35 EXAMPLE The process is not complete! To obtain a cohesive, connected object, repeat the procedure on the WHITE pixels Complement the last result: 35

36 EXAMPLE Then apply all the same steps: "Color" of largest blob: 1 36

37 EXAMPLE Simple and effective, but doesn t "cure" everything 37

38 BINARY MORPHOLOGY The most powerful class of binary image operators A general framework known as mathematical morphology morphology = shape Morphological operations affect the shapes of objects and regions in binary images All processing is done on a local basis - region or blob shapes are affected in a local manner Morphological operators Expand (dilate) objects Shrink (erode) objects Smooth object boundaries and eliminate small regions or holes Fill gaps and eliminate peninsulas All is accomplished using local logical operations 38

39 STRUCTURING ELEMENTS OR WINDOWS A structuring element is a geometric relationship between pixels Some examples: Morphological operations are defined (conceptually) by moving a structuring element over the image to be modified, in such a way that it is centered over every image pixel at some point 39

40 STRUCTURING ELEMENTS Usually this is done row-by-row, column-by-column When the structuring element is centered over a region of the image, a logical operation is performed on the pixels covered by the structuring element, yielding a binary output A structuring element is also often called a moving window Usually structuring elements are defined to have (approximate) circular shapes - since it is desired that they interact the same way with an object even if the object is rotated 40

41 EXAMPLE 41

42 FORMAL DEFINITION OF WINDOWING Also used later for gray-level image processing A window is a geometric relationship that creates a series of miniature images as it is passed over the image, row-by-row, columnby-column (sequential implementation) In a parallel implementation, a large number of windows will cover the image simultaneously 42

43 Some typical windows: WINDOWING These operate on rows and columns only A window will always cover an odd number of pixels 2M+1: pairs of adjacent pixels, plus the center pixel Filtering operations are defined symmetrically this way 43

44 TWO-DIMENSIONAL WINDOWS Again, 2M+1 denotes the odd number of pixels covered by the window Can generalize to arbitrary-size windows covering 2M+1 pixels These are the most common window shapes 44

45 WINDOW NOTATION A window B is: A way of collecting local image intensities. A set of coordinate shifts B i = (m i, n i ) centered around (0, 0): B = {B 1,..., B 2M+1 } = {(m 1, n 1 ),..., (m 2M+1, n 2M+1 )} 45

46 EXAMPLES - 1-D WINDOWS B 46

47 EXAMPLES - 2-D WINDOWS B 47

48 WINDOWED SET Given an image I and a window B, define the windowed set at image coordinate (i, j) by B I(i, j) = {I(i-m, j-n); (m, n) B} which is the set of image pixels covered by the window when it is centered at coordinate (i, j) B = ROW(3): B I(i, j) = {I(i, j-1), I(i, j), I(i, j+1)} 48

49 EXAMPLES 49

50 GENERAL BINARY FILTER Denote binary operation G on the windowed set B I(i, j) by: J(i, j) = G{B I(i, j)} = G{I(i-m, j-n); (m, n) B} Perform this at every pixel in the image, giving filtered image J = G[I, B] = [J(i, j); 0 <= i, j <= N-1] 50

51 EDGE-OF-IMAGE PROCESSING Window overlapping "empty space" : Convention: fill the "empty" window slots by the nearest image pixel. This is called replication 51

52 DILATION, EROSION AND MEDIAN (MAJORITY) DILATION: Given a window B and a binary image I: J 1 = DILATE(I, B) if J 1 (i, j) = OR{B I(i, j)} = OR{I(i-m, j- n); (m, n) B} EROSION: Given a window B and a binary image I: J 2 = ERODE(I, B) if J 2 (i, j) = AND{B I(i, j)} = AND{I(i-m, j-n); (m, n) B} MEDIAN: Given a window B and a binary image I: J 3 = MEDIAN(I, B) if J 3 (i, j) = MAJ{B I(i, j)} = MAJ{I(i-m, j-n); (m, n) B} 52

53 DILATION So-called because this operation increases the size of BLACK objects in a binary image Local Computation: J = DILATE(I, B) 53

54 DILATION So-called because this operation increases the size of BLACK objects in a binary image Local Computation: J = DILATE(I, B) 54

55 DILATION Global Effect: 55

56 EROSION So-called because this operation decreases the size of BLACK objects in a binary image Local Computation: J = ERODE(I, B) 56

57 EROSION So-called because this operation decreases the size of BLACK objects in a binary image Local Computation: J = ERODE(I, B) 57

58 EROSION Global Effect: 58

59 QUALITATIVE PROPERTIES OF DILATION Dilation removes object holes of too-small size: Dilation also removes gaps or bays of toonarrow width: 59

60 QUALITATIVE PROPERTIES OF DILATION Dilation of the BLACK part of an image is the same as erosion of the WHITE part! 60

61 QUALITATIVE PROPERTIES OF EROSION Erosion removes objects of too-small size: Erosion also removes peninsulas of too-narrow width: 61

62 QUALITATIVE PROPERTIES OF EROSION Erosion of the BLACK part of an image is the same as dilation of the WHITE part! 62

63 RELATING EROSION AND DILATION Erosion and dilation are actually the same operation - they are just dual operations with respect to complementation Erosion and dilation are only approximate inverses of one another Dilating an eroded image rarely yields the original image In particular, dilation cannot Recreate peninsulas eliminated by erosion Recreate small objects eliminated by erosion Eroding a dilated image rarely yields the original image In particular, erosion cannot Unfill holes filled by dilation Recreate gaps or bays filled by dilation 63

64 MEDIAN Actually majority. A special case of the gray-level median filter Possesses qualitative attributes of both dilation and erosion, but does not generally change the size of objects or background Local Computation: J = MEDIAN(I, B) The median removed the small object A and the small hole B, but did not change the boundary (size) of the larger region C 64

65 QUALITATIVE PROPERTIES OF MEDIAN Median removes both objects and holes of too-small size, as well as both gaps (bays) and peninsulas of too-narrow width 65

66 QUALITATIVE PROPERTIES OF MEDIAN Note that median does not generally change the size of objects (although it does alter them) Median is its own dual, since MEDIAN [ NOT(I) ] = NOT [ MEDIAN(I) ] Thus, the median is a shape smoother. It is a filter We can define other shape smoothers as well. 66